Split evenly: divide a city into areas that hold the same amount

Nine areas of Montreal, each with the same number of places to eat and drink

Say a food and drink supplier has nine sales reps in Montreal and wants to give each one an equal part of town - the same number of restaurants, cafés, bars and bakeries to call on. Cut the map into nine equal-looking slices and one rep gets downtown, with more places in a few streets than another rep has in a whole suburb. Split evenly draws the nine areas from the data instead: you pick what to balance and how many areas, and it divides the city into joined-up areas that each hold about the same amount.

Inside Montreal's city boundary it found 8,076 places to eat and drink and made nine areas of 895 to 904 each - every one within 0.7% of the even share of 897. The smallest areas, around downtown and the Plateau, need fewer than 100 hexes to get there; the one on the West Island needs 1,984. Same number of places, an area 21 times larger.

Split evenlythe field's term: a capacity-constrained power diagram, polished

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Why split a city evenly?

Whenever a city is shared between people - reps, drivers, crews, franchisees, volunteers - the fair way to share it is by what each part holds, not by how big it looks on the map. A few everyday jobs it does:

Want to know which of your stores each hex can get to soonest instead, whatever that does to the totals? That is the Divide tool's other section, Closest.

How to use it, step by step

Split evenly is in the Divide tool, beside Explore, Territory and Market Share in the tool menu: pick Divide, then Split evenly in the menu right next to it. While it is open, the map shows only the split.

1. Pick what to balance. Press Change measures… and pick from the same catalog the rest of MapBees uses. Only measures that add up are offered - a count of places, residents, building volume, your own data - because an average or a share can't be split.
2. Say where. Out of the box, Split evenly divides the box drawn around the city. To split the city itself, press Draw or pick an area on the map… and pick the city's boundary - Montreal (Montréal) - or one neighborhood, or draw your own shape.
3. Say how many areas. 2 to 24, or switch to About … per area and type how much each area should hold; MapBees works out how many areas that makes.
4. Press Split evenly. The areas draw on the map in about a second, numbered, with touching areas never sharing a color, and the list under the settings fills in.

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Every setting, in plain words

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• Balance on - the measure every area should hold the same amount of. The first is the main one; up to 3 more can also be kept even, though one splits best (more on that below). In this example: Eat & Drink.
• Split inside - what gets divided: the box drawn around the city (the starting choice), the city's own boundary or a neighborhood picked on the map, a shape you draw, or one of your own areas. In this example: Montreal (Montréal), the city boundary.
• Start from - where the areas grow from. Automatic picks the starting hexes for you; On the map lets you click one starting hex per area; Your points uses a file of your own points, such as your stores; My areas rebalances areas you already have. In this example: Automatic.
• How many - a number of areas from 2 to 24, or About … per area. In this example: 9 areas.
• Allowed gap - how far an area may sit from the even share and still count as even: 5, 10 or 15%. In this example: 10%.
• Roundness - how much round areas matter next to evenness: Any, Rounder or Round. Rounder areas have shorter borders and are easier to drive. In this example: Rounder.
• Hexes with no data - what happens to a hex the measure has no figure for: it joins an area (adding nothing) or is left out of every area. In this example: Leave them out.
• Show 3 ways to split - makes three different splits of the same request to pick from.

Reading the result

The top of the list says whether the split worked - here a green All 9 areas within 10% - then the total and the even share: Eat & Drink: 8,076 in all · even share 897 per area. Each area gets a row with a meter, its amount and share of the total, its number of hexes and its roundness.

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On the map, every hex takes its area's color, deeper the more of the measure it holds, on one scale for the whole split - so a deep hex downtown and a deep hex in Laval hold about the same. Hover any hex for its area and its own amount.

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The note One hex holds 11% of an even share of Eat & Drink is the panel being honest about a lumpy measure. Downtown, a single hex holds about a hundred places, and a hex can't be cut in half - so an area can only be as even as its biggest hexes allow. Here it still worked: every area landed within 0.7%. With more areas, or a stricter allowed gap, the same hex can push one past the edge, and the panel warns before that happens.

The even share, worked through

3 ways to split, and Another way

There is rarely one right split. Tick Show 3 ways to split and press Split again: three different splits of the same request come back side by side, each a small map with its biggest gap and how round its areas are. Pick the one that suits you - say, the one that keeps a neighborhood your rep already knows in one area. Another way makes the next one.

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Watch the steps

Watch the steps, under the list, makes the same split again on the map, slowly, in its three steps: starting hexes, grow, polish. It is the quickest way to see how the method works.

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The math, honestly

For anyone who wants the working: Split evenly is a balanced partition of the hex map, a problem known in the field as territory design. Each hex is about 350 m across (H3 resolution 9), and two hexes are neighbors when they share an edge. It runs in three steps.

1. Starting hexes, with k-means++. One starting hex per area is picked where the measure is, not at random: each next one is drawn with a chance that grows with its distance from those already picked and with how much of the measure is there - the k-means++ rule (Arthur and Vassilvitskii, 2007). A few rounds of Lloyd's algorithm then move each one toward the middle of what it gathers.
2. Grow, as a capacity-constrained power diagram. Every hex goes to the area for which distance² − reach is smallest. An area holding less than its even share gets more reach and pulls hexes in along its whole border at once; one holding too much gets less and gives hexes back. The reaches are adjusted until every area holds about its share. This is a power diagram - a Voronoi diagram in which each area's middle carries a weight, also called a Laguerre diagram - balanced to equal amounts (Aurenhammer, Hoffmann and Aronov, 1998). Its borders are straight, so areas come out without lobes or tentacles, and a few Lloyd rounds move each area's middle to the middle of its hexes to even the outlines out.
3. Polish, by local search. A hex on a border moves to a touching area whenever that brings the totals closer to even or shortens the border; Roundness sets how much a border costs. A move that would cut the area it leaves in two, or leave a hex hanging by one edge, is never made, so every area stays in one piece - the contiguity rule of graph partitioning. On Montreal the polish made 51 moves.

• Roundness on each row compares an area's spread with a disc's - the normalized moment of inertia. A disc scores 100%; a long or ragged area scores less.
• Same request, same split. The method has no hidden randomness, so a shared link makes exactly the same areas on arrival. The 3 ways start from different starting hexes; ways that differ in fewer than 2% of hexes are dropped, and the rest are ranked: every area within the allowed gap first, then the smallest biggest gap, then the roundest.
• Several measures. With more than one measure kept even, a second attempt runs beside the first in which each area also carries a price per measure - Lagrange multipliers, in the field's terms - and the better of the two is kept.

One measure splits best

You can keep up to four measures even at once - say Eat & Drink and residents. The panel tells you straight away why that is harder: each extra measure is one more total every area has to match.

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In Montreal the two pull apart. Downtown has far more places to eat and drink per resident than the suburbs, so an area that holds its share of places holds too few residents, and the other way round. Asked for both at a 10% allowed gap, Split evenly keeps residents just inside (about ±9.8%) but 7 of the 9 areas miss on Eat & Drink, by up to 14.8% - and the areas come out ragged: 60% round on average, against 83% with Eat & Drink alone.

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Take the areas with you

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• Copy to Areas puts copies of the areas in the Territory tool, where each gets the full dashboard of residents, places and buildings - see the Territory mapping guide.
• GeoJSON downloads one shape per area with its name, number, totals and gap from even; CSV downloads one row per hex: h3,area,area_name - ready for a CRM or a routing tool.
• Share the split with the link button: the link carries the recipe, and the same split is made again when it opens.
• Already have areas? Start from: My areas checks how even your own areas are (Only check), moves as few border hexes as it can to even them out (Rebalance), or splits them afresh (Start over). Locked areas keep exactly their hexes.
• Hand-tune with a brush moves hexes between areas by hand with live meters; Rebalance the rest then evens out everything you didn't touch.
• Your own data - a file of your points can start the areas, or a count or sum from it can be what is balanced. See Your data.

What Split evenly can't tell you

Split Montreal into 9 even areas

The whole idea, in eight lines

• Split evenly divides a city, one of your areas or your own areas into 2 to 24 areas that each hold about the same amount.
• The even share is the total divided by the number of areas; the allowed gap (5, 10 or 15%) says how close counts as even.
• Balance on anything that adds up: places, residents, building volume or your own data.
• Montreal, inside its city boundary, Eat & Drink, 9 areas: 895 to 904 places each - 94 hexes at the smallest, 1,984 on the West Island.
• Starting hexes by k-means++, areas grown as a capacity-constrained power diagram, then a polish that keeps every area in one piece.
• 3 ways to split and Another way give different, equally even splits to pick from.
• One measure splits best; combine measures that belong together.
• Copy to Areas, download GeoJSON or CSV, or share a link that makes the same split again.